
Taking graphs with respect to the decompositions \(V\oplus V^*\)  resp. \(V\oplus \mathrm {graph}(Z)\) , yields bijections

02 V*  {Lagrangian subspace of \( transverse to \)  V*},
  {(v,v) | vV},
Z2 V*  {Lagrangian subspace of \( transverse to \)  graph(Z)},
{(v+Z♯(v),v) | vV}.
\((ii)\)
Given \(\beta \in \wedge ^2 V^*\) , the Lagrangian subspace \(\Phi _Z(\beta )\)  is transverse to \(V^*\subset  if, and only if {\beta \in \mathcal {I}_Z}.\item [\) (iii)\(]The map \begin{gather*}\Phi _0^{-1}\circ \Phi _Z\colon \ \mathcal {I}_Z\rightarrow \wedge ^2 V^*\end{gather*} is well-defined and coincides with~\) F\(.\)
In particular, the map \(F\)  is characterized by the property that
\(\mathrm {graph}(F(\beta ))=\Phi _Z(\beta )\) 
for all \(\beta \in \mathcal {I}_Z\) . In other words, \(F(\beta )\)  is obtained taking the graph of \(\beta \)  w.r.t. the splitting \(V\oplus \mathrm {graph}(Z)\) .
(i) According to Remark REF , any Lagrangian subspace transverse to \(V^*\)  is the graph of a skew-symmetric linear map \(V\rightarrow V^*\) , and therefore can be written as \(\lbrace (v,\iota _v\alpha )\,|\,v\in V\rbrace \)  for some \(\alpha \in \wedge ^2 V^*\) . Similarly, \(\mathrm {graph}(Z)\)  is transverse to \(V\)  and the induced isomorphism \(\mathrm {graph}(Z)\cong V^*\)  is just \((Z^\sharp (\xi ),\xi )\mapsto \xi \) . Hence any Lagrangian subspace transverse to \(\mathrm {graph}(Z)\)  can be written as \(\lbrace (v,0)+(Z^\sharp (\iota _v\beta ),\iota _v\beta )\,|\,v\in V\rbrace \)  for some \(\beta \in \wedge ^2 V^*\) .
(ii) The expression for \(\Phi _Z(\beta )\)  in item (i) shows that \(\Phi _Z(\beta )\cap V^* =\lbrace (0,\iota _v\beta )\,|\,v\in V, v+Z^\sharp (\iota _v\beta )=0\rbrace \) . This intersection is trivial iff \(\ker \big (\mathrm {id} + Z^\sharp \beta ^\sharp \big ) \subseteq \ker \big (\beta ^\sharp \big )\) . In turn, this condition is equivalent to \(\big (\mathrm {id} + Z^\sharp \beta ^\sharp \big )\)  being injective, and thus invertible.
(iii) Finally, if \(\mathrm {id}+Z^\sharp \beta ^\sharp \)  is invertible, \(\Phi _Z(\beta )\)  is transverse to \(V^*\)  by item (ii). By item (i) the element \(\Phi _0^{-1}(\Phi _Z(\beta ))\)  is well-defined. In concrete terms, it is given by \(\alpha \in \wedge ^2 V^*\)  such that for all \(v\in V\) , there is \(w \in V\)  for which
\(\big (v + Z^\sharp \beta ^\sharp (v),\beta ^\sharp (v)\big )=\big (w,\alpha ^\sharp (w)\big )\) 
holds. Equivalently, this means that \(\alpha ^\sharp \big (\mathrm {id}+Z^\sharp \beta ^\sharp \big )(v)=\beta ^\sharp (v)\)  for all \(v\in V\) . This shows that \(\Phi _0^{-1}\circ \Phi _Z\)  agrees with \(F\) .
Revisiting Theorem REF  (parametrizing constant rank forms)
Now let \(\eta \in \wedge ^2 V^*\)  be of rank \(k\) , fix a complement \(G\)  to \(K:=\ker (\eta )\) , and denote by \(Z \in \wedge ^2 G\)  the bivector determined by \( Z^\sharp =-\big (\eta \vert _G^\sharp \big )^{-1}\) . In Section REF  we considered deformations of the Dirac structure \((\eta )\)  using \(G\oplus K^*\)  as a complement. They are graphs of 2-forms given by the Dirac exponential map \(\exp _{\eta }\)  (see Definition REF ). More precisely:
Lemma 3.10  For all \(\beta \in \mathcal {I}_Z\)  we have
\(\mathrm {graph}(\exp _{\eta }(\beta ))=\Phi _{G\oplus K^*}\big (\bar{\beta }\big ).\) 

We have \(\mathrm {graph}(\exp _{\eta }(\beta ))=_\eta (\Phi _Z(\beta ))=\Phi _{G\oplus K^*}\big (\bar{\beta }\big )\) , where the first equality holds by equation (REF ) and the second by Lemma REF .
Using this we recover Theorem REF , in particular item (i) stating that \(\exp _{\eta }(\beta )\)  has rank equal to \(k=\dim (K)\)  iff \(\beta \)  is horizontal.
[Alternative proof of Theorem REF ] (i) Apply Proposition REF (ii) together with equation (REF ).
(ii) We only prove the statement about the kernel of \(\exp _{\eta }(\beta )\) . Write \(\beta = (\mu ,\sigma )\) . By the proof of Proposition REF (i), the intersection of the subspace (REF ) with \(V\)  is \((_{\eta }\circ _{Z})(\mathrm {graph}(\beta )\cap (K\oplus G^*))\) , which is precisely the image of \(K\)  under \(\mathrm {id}+ Z^\sharp \mu ^\sharp \) .
(iii) By Lemma REF (ii), the map \(\Phi _Z\)  provides a bijection between \(\mathcal {I}_Z\)  and Lagrangian subspaces transverse to \({\mathrm {graph}(Z)}\)  and to \(V^*\) . Hence \(_\eta \circ \Phi _Z\) 
provides a bijection between \(\mathcal {I}_Z\)  and Lagrangian subspaces transverse to \(_\eta ({\mathrm {graph}(Z)})=G\oplus K^*\)  (see Lemma REF ) and to \(V^*\) . The latter are exactly the graphs of elements \(\eta ^{\prime } \in \wedge ^2 V^*\)  so that the \(\eta ^{\prime }|_{\wedge ^2G}\)  is non-degenerate. Hence, by the proof of Lemma REF , \(\exp _{\eta }\)  provides a bijection between \(\mathcal {I}_Z\)  and such \(\eta ^{\prime }\) . We conclude using (i).

Dirac-geometric interpretation of Section  REF
Using Dirac geometry and adapting results from [1]}, we explain and re-prove the results recalled in Section REF . Fix a bivector field \(Z\)  on \(M\) .
Revisiting Proposition REF  (the \({L_\infty [1]}\) -algebra \({\Omega (M)[2]}\) )Revisiting Proposition REF  (the \(L_\infty [1]\) -algebra \(\Omega (M)[2]\) )
In Proposition REF , the \(L_\infty [1]\) -algebra \((\Omega (M)[2],\lambda _1,\lambda _2,\lambda _3)\)  was constructed out of a bivector field \(Z\) . It can be recovered using Dirac geometry – or more precisely, the deformation theory of Dirac structures – as a special case of the construction from [1]}.
Proposition 3.11  Let \(L\)  be a Dirac structure and \(R\)  a complementary almost Dirac structure, i.e., we have a vector bundle decomposition \(L\oplus R=\mathbb {T}M\) . Then \(\Gamma (\wedge L^*)[2]\)  has an induced \(L_{\infty }[1]\) -algebra structure, whose only non-trivial multibrackets are \(\mu _1\) , \(\mu _2\) , \(\mu _3\)  given as follows:

\(\mu _1\)  is the differential \(d_L\)  associated to the Lie algebroid \(L\) ,

\(\mu _2(\alpha [2] \odot \beta [2])=-(-1)^{|\alpha |}[\alpha ,\beta ]_{L^*}[2]\) , where \([\cdot ,\cdot ]_{L^*}:=\mathrm {pr}_{R}([\![\cdot , \cdot ]\!])\)  denotes the \((\) extension of\()\)  the bracket of the almost Lie algebroid \(R\cong L^*\) ,

\(\mu _3(\alpha [2]\odot \beta [2]\odot \gamma [2])=(-1)^{|\beta |} \big (\alpha ^\sharp \wedge \beta ^\sharp \wedge \gamma ^\sharp \big ) \psi [2]\) , where \(\psi \in \Gamma \big ({\wedge }^3 L\big )\)  is given by \(\Gamma \big ({\wedge }^3 L^*\big )\rightarrow \mathcal {C}^\infty (M)\) , \(\xi _1 \wedge \xi _2\wedge \xi _3 \mapsto \langle \mathrm {pr}_{L}([\![\xi _1 , \xi _2 ]\!]), \xi _3\rangle \) , where we made use of the identification \(R \cong L^*\) .


More generally, Proposition REF  holds if replacing \(\mathbb {T}M\)  by any Courant algebroid.
The proof is a minor adaptation of the first part of the proof of [1]}, setting \(\varphi =0\)  there. We recall briefly the idea of the latter. By [4]} there is a natural description of the Courant algebroid structure on \(\mathbb {T}M\)  in terms of graded geometry. One can use it to apply Voronov's higher derived brackets construction (see [5]}, [6]}) and obtain an \(L_{\infty }[1]\) -algebra structure on \(\Gamma (\wedge L^*)[2]\) . The multibrackets obtained are the ones in the statement of the lemma, as one checks using [4]} and via computations in local coordinates.
[Alternative proof of Proposition REF ] Let \(Z\)  be a bivector field on \(M\) . We apply Proposition REF  for the case \(L=TM\)  and \(R=\mathrm {graph}(Z)\) . In this case \(d_L\)  is the de Rham differential, and the bracket on \(R\)  is given by the formula for the Koszul bracket. One checks that \(\psi \)  is the trivector field \(-\frac{1}{2}[Z,Z]\) , using [8]}. Hence the \(L_{\infty }[1]\) -brackets on \(\Omega (M)[2]\)  given by Proposition REF  are \(\mu _1 =\lambda _1\) , \(\mu _2 =-\lambda _2\)  and \(\mu _3 =\lambda _3\) . Applying the automorphism \(-\mathrm {id}\)  to \(\Omega (M)[2]\)  yields Proposition REF .

Revisiting Corollary REF  (Maurer–Cartan elements of \({\Omega (M)[2]}\) )Revisiting Corollary REF  (Maurer–Cartan elements of \(\Omega (M)[2]\) )
We now turn to Maurer–Cartan elements. In Lemma REF (i), we gave a parametrization of all almost Dirac structures that are transverse to \(\mathrm {graph}(Z)\)  in terms of 2-forms \(\beta \)  on \(M\) . This parametrization is given by
\(\beta \mapsto \Phi _Z(\beta )=\big \lbrace \big (v+Z^\sharp (\iota _v\beta ),\iota _v\beta \big ) \, | \, v \in TM\big \rbrace .\) 
We present the second part of [1]}, which is an extension of the work by Liu–Weinstein–Xu recalled in Remark REF .
Proposition 3.12 Let \(L\)  be given a Dirac structure and \(R\)  a complementary almost Dirac structure. An element \(\sigma \in \Gamma \big ({\wedge }^2 L^*\big )[2]\)  is a Maurer–Cartan element of the \(L_{\infty }[1]\) -algebra structure given in Proposition REF  iff the graph
\(\Gamma _{\sigma }:= \lbrace  (X-\iota _X \sigma )\colon X\in L\rbrace \subset L\oplus R\) 
is a Dirac structure. \((\) The above inclusion makes use of the identification \(R\cong L^*\) .\()\)

Corollary REF  states that for \(\beta \in \Omega ^2(M)\)  taking values in some sufficiently small neighborhood \(\mathcal {U}\)  of the zero section in \(\wedge ^2 T^*M\)  – in particular taking values in \(\mathcal {I}_Z\) , i.e., \(\mathrm {id} + Z^\sharp \beta ^\sharp \)  is invertible, – \(\beta \)  is a Maurer–Cartan element of \((\Omega (M)[2],\lambda _1,\lambda _2,\lambda _3)\)  iff \(F(\beta )\) 
is closed. We now provide an alternative proof of this result, which also shows that one can choose \(\mathcal {U}\)  to equal \(\mathcal {I}_Z\) .
[Alternative proof of Corollary REF ] For any \(\beta \in \Omega ^2(M)\) , being a Maurer–Cartan element of the \(L_{\infty }[1]\) -algebra \((\Omega (M)[2],\lambda _1,\lambda _2,\lambda _3)\)  is equivalent to \(\Phi _Z(\beta )\)  being a Dirac structure. This follows from applying Proposition REF  to the Dirac structure \(L=TM\)  and to the almost Dirac structure \(R=\mathrm {graph}(Z)\) , noticing that \(\Gamma _{-\beta }=\lbrace (v+Z^\sharp (\iota _v\beta ),\iota _v\beta )\,|\,v\in TM\rbrace =\Phi _Z(\beta )\) . When \(\beta \in \Gamma (\mathcal {I}_Z)\) , we know that \(\Phi _Z(\beta )\)  can be written as the graph of the 2-form \(F(\beta )\) , by equation (REF ). Now use the fact that the graph of a 2-form is a Dirac structure if, and only if, the 2-form is closed.
Remark 3.13 In this subsection we recovered the \(L_\infty [1]\) -algebra \(\Omega (M)[2]\)  of Proposition REF  as the \(L_\infty [1]\) -algebra governing deformations of the Dirac structure \(TM\)  taking \((Z)\)  as a complement. By Remark REF , this deformation problem is equivalent to the deformations of the Dirac structure \(\mathrm {graph}(\eta )\)  taking \(G\oplus K^*\)  as the complement. This explains why the \(L_\infty [1]\) -algebra \(\Omega (M)[2]\)  governs the latter deformation problem, and therefore is relevant for the deformations of pre-symplectic structures.


Dirac-geometric interpretation of Section  REF
Theorem REF  can be deduced from a general statement about (almost) Dirac structures, however doing so amounts essentially to the same computations that were needed for the proof given in [8]}. We include this general statement for the sake of completeness.
Proposition 3.14 In the setting of Proposition REF , let \(K\)  be a subbundle of \(L\)  and define \(\Gamma _{\mathrm {hor}}(\wedge L^*)\)  as the kernel of the restriction map \(\Gamma (\wedge L^*)\rightarrow \Gamma (\wedge K^*)\) . Then the multibrackets \(\mu _1\) , \(\mu _2\) , \(\mu _3\)  preserve \(\Gamma _{\mathrm {hor}}(\wedge L^*)[2]\)  iff \(K\)  satisfies the following:

\(K\)  is a Lie subalgebroid of \(L\) ,

\(\langle [\![\xi _1 , \xi _2 ]\!],\, K+K^{\circ } \rangle =0\)  for all \(\xi _1 , \xi _2 \in \Gamma (K^{\circ })\) , where we use the identification \(K^{\circ }\subset L^*\cong R\)  and \([\![\cdot ,\cdot ]\!]\)  denotes the Dorfman bracket.


We will use the fact that \(\mu _1\) , \(\mu _2\) , \(\mu _3\)  are derivations w.r.t. the wedge product in each entry. The Lie algebroid differential \(d_L\)  preserves \(\Gamma _{\mathrm {hor}}(\wedge L^*) \)  iff the subbundle \(K\)  is involutive. The bracket \([\cdot ,\cdot ]_{L^*}\)  preserves \(\Gamma _{\mathrm {hor}}(\wedge L^*) \)  iff \(\langle [\![\xi _1 , \xi _2 ]\!], K\rangle =0\)  for all \(\xi _1 , \xi _2 \in \Gamma (K^{\circ })\) . The trinary bracket \(\mu _3\)  preserves \(\Gamma _{\mathrm {hor}}(\wedge L^*) \)  iff \(\mu _3(\xi _1,\xi _2,\xi _3)=0\)  for all \(\xi _i \in \Gamma (K^{\circ })\) , which in turn is equivalent to \(\langle [\![\xi _1 , \xi _2 ]\!], \xi _3 \rangle =0\) .
Finally, as mentioned earlier, Theorem REF  follows immediately from the other results presented.

Acknowledgements
We thank Stephane Geudens for comments on a draft of this note. M.Z. acknowledges partial support by IAP Dygest, the long term structural funding – Methusalem grant of the Flemish Government,
the FWO under EOS project G0H4518N, the FWO research project G083118N (Belgium).
[1]Referencesref
